Frameworks for Two-Dimensional Keller Maps

The classical Jacobian Conjecture asserts that every locally invertible polynomial self-map of the complex affine space is globally invertible. A Keller map is a (hypothetical) counterexample to the Jacobian Conjecture. In dimension two every such map, if exists, leads to a map between the Picard groups of suitable compactifications of the affine plane, that satisfy a complicated set of conditions. This is essentially a combinatorial problem. Several solutions to it ("frameworks") are described in detail. Each framework corresponds to a large system of equations, whose solution would lead to a Keller map.

An ideal-theoretic approach to Keller maps

A self-map F of an affine space ${\bf A}_k^n $ over a field k is said to be a Keller map if F is given by polynomials F1, …, Fn ∈ k[X1, …, Xn] whose Jacobian determinant lies in $k\setminus \{0\}$. We consider char(k) = 0 and assume, as we may, that the Fis vanish at the origin. In this note, we prove that if F is Keller then its base ideal IF = 〈F1, …, Fn〉 is radical (a finite intersection of maximal ideals in this case). We then conjecture that IF = 〈X1, …, Xn〉, which we show to be equivalent to the classical Jacobian Conjecture. In addition, among other remarks, we observe that every complex Keller map admits a well-defined multidimensional global residue function.

CLASSIFICATION OF CUBIC HOMOGENEOUS POLYNOMIAL MAPS WITH JACOBIAN MATRICES OF RANK TWO

Let $K$ be any field with $\text{char}\,K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ whose Jacobian matrix, ${\mathcal{J}}H$, has $\text{rk}\,{\mathcal{J}}H\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map, then $F$ is invertible and furthermore $F$ is tame if the dimension $n\neq 4$.

Irreducibility Properties of Keller Maps

Jędrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducibility properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over ℂ and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.